Robust Wirtinger Flow for Phase Retrieval with Arbitrary Corruption

We consider the robust phase retrieval problem of recovering the unknown signal from the magnitude-only measurements, where the measurements can be contaminated by both sparse arbitrary corruption and bounded random noise. We propose a new nonconvex algorithm for robust phase retrieval, namely Robust Wirtinger Flow to jointly estimate the unknown signal and the sparse corruption. We show that our proposed algorithm is guaranteed to converge linearly to the unknown true signal up to a minimax optimal statistical precision in such a challenging setting. Compared with existing robust phase retrieval methods, we achieve an optimal sample complexity of $O(n)$ in both noisy and noise-free settings. Thorough experiments on both synthetic and real datasets corroborate our theory.